Papers

For the mass-critical generalized Korteweg-de Vries equation, \begin{equation*} \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}, \end{equation*} we prove the existence of three-soliton solutions with logarithmic relative distance and with the choice of signs $(+,-,-)$. The choice of the number three and the signs of solitons are related to the solvability of the ODE system generated by the nonlinear interactions between the three solitons and some non-localized profiles. In particular, these special behaviors are due to strong interactions between the three solitons. That is, the dynamics of each soliton is perturbed at leading order by the presence of other solitons.

We develop a constructive method for solving semilinear elliptic equations $\Delta u(x)=f(x,u(x))$ on locally finite, connected infinite graphs with layered structure. Using Eidelheit's theorem, we establish coupling criteria ensuring that arbitrary initial-layer data extend to global solutions for every $f$. We apply combinatorial criteria to prove solvability on leafless infinite trees, integer lattices, the triangular and hexagonal lattices, and a Cayley graph of the discrete Heisenberg group. We further establish solvability for a broad class of Cayley graphs of semidirect products $G\cong\mathbb Z\ltimes_\theta H$. In particular, $\Delta u=e^u$ has infinitely many solutions on $\mathbb Z^2$, but none of finite energy. We also extend the method to the bi-Laplacian under two-step coupling conditions, to the $p$-Laplacian under a unique-neighbor condition, and to magnetic Laplacians.

Layer potentials provide a classical approach to boundary value problems for Laplace's equation on Lipschitz domains. Kenig's 1994 spectral-radius conjecture for the double-layer operator would ensure operator-norm convergence of the associated Neumann series on mean-zero $L^2$ densities when the boundary is connected. We disprove this conjecture by constructing a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. More precisely, for every $t>1/2$ sufficiently close to $1/2$, we obtain such a domain with $\pm i t$ in its Fredholm essential spectrum. The construction starts from smooth graphs whose shapes repeat under translation. In the limit of separated scales, refinement makes solutions of adjoint resolvent equations grow with fixed forcing. The graph slopes remain uniformly bounded. A computer-assisted certificate proves this growth through an inequality for Hermitian $2\times2$ matrices. Its strict margin at $- i/2$ persists at nearby spectral parameters. Normalisation and a Floquet transform then give compactly supported densities with small residuals on the full graphs. We insert rescaled segments of successive graphs into one bounded boundary, where these densities form a weakly null sequence of approximate eigenvectors. The same spectral conclusion holds on a single periodic Lipschitz graph. The certificate combines continuous estimates, exact rational arithmetic and rigorous interval enclosures.

DNA replication requires the coordination of origin firing and fork progression to ensure the entire genome is timely duplicated before cell division. Yet origin firing is stochastic, giving rise to the classical random completion problem of how probabilistic local events can nevertheless ensure reliable genome duplication. Although several biological mechanisms have been proposed to resolve this problem, a quantitative account of how heterogeneous initiation and fork speed govern the persistence of the final unreplicated regions is still lacking. To address this gap, we introduce a population-level kinetic framework that extends KJMA nucleation-and-growth models by tracking unreplicated intervals over size, genomic position and time. We establish well-posedness of the resulting mean-field system and global existence for compatible data. Notably, by introducing a local initiation mass function, we quantify how the density and spatial organisation of origin firing constrain replication completion, yielding novel and sharp upper bounds on both the worst-locus unreplicated fraction and locuswise near-completion time. These results provide a rigorous and computable foundation for mapping vulnerabilities in replication completion and relating persistent unreplicated regions to replication stress and genome instability.

This monograph presents a systematic account of several classical and recent developments in the theory of rotating vortex patches, or V-states, for the two-dimensional Euler equations and related active scalar models. Its purpose is both expository and structural: we revisit some of the foundational results of the subject, provide detailed proofs and alternative formulations, and present more recent results within a common analytical framework. We begin with Euler vortex patches and the contour dynamics formulation of rigidly rotating solutions. We then develop two complementary approaches to the V-state equation, based on Cauchy integrals and conformal mappings, and discuss their connections with potential theory and Faber polynomials. These tools are used to revisit classical examples, including Rankine vortices and Kirchhoff ellipses, as well as rigidity and classification results. A substantial part is devoted to Burbea construction of noncircular V-states via bifurcation theory. The final part develops a unified approach to rotating patches for a broad class of incompressible active scalar equations. Structural properties of the interaction kernel, in particular complete monotonicity and the resulting spectral factorization, yield a common bifurcation framework encompassing the Euler, generalized surface quasi-geostrophic, quasi-geostrophic shallow-water, and related models.

We study the initial-boundary value problem for the Short Wave equation on the half-line $x\ge 0$. A distinctive feature of this problem is that the boundary $x=0$ cannot be characterized a priori as an inflow or outflow boundary: its character is determined dynamically by the sign of the unknown trace $u(0,t)$. This leads to different analyticity properties of the associated eigenfunctions and, consequently, to different spectral formulations in the regimes $u(0,t)\le0$ and $u(0,t)\ge0$. Using the Unified Transform Method (aka the Fokas method), we formulate the solution in terms of matrix Riemann–Hilbert problems. We construct the associated spectral functions, derive the global relations, and show how the solution is reconstructed from the corresponding Riemann–Hilbert problem. In the case $u(0,t)\le0$, the solution is determined by the initial data alone (assuming an appropriate decay as $x\to \infty$), whereas for $u(0,t)\ge0$, compatible boundary data are also required for the construction.

We consider positive weak solutions of $-\Delta_p u=f(u)$, with $f$ positive and locally Lipschitz continuous. In the singular case $1<p<2$, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Damascelli and Sciunzi [Calc.\ Var.\ Partial Differential Equations 25 (2006), 139–159] have been available, for more than twenty years, only under the restriction $\frac{2N+2}{N+2}<p<2$, a range that shrinks to the empty set as $N\to\infty$. We remove the dependence on the dimension and prove all of these results for $\frac32<p<2$. Since the strong comparison principle is used as a black box throughout the qualitative theory of $p$-Laplace equations, the improvement propagates. We show in particular that the assumption $\frac{2N+2}{N+2}<p<2$ may be replaced by $\frac32<p<2$ in the resolution of Gibbons' conjecture for $-\Delta_pu=f(u)$ by Esposito, Farina, Montoro and Sciunzi [Math.\ Ann.\ 382 (2022), 943–974] and in their monotonicity theorem in half-spaces for changing-sign nonlinearities [Calc.\ Var.\ Partial Differential Equations 61 (2022), art.\ 154]; further applications are given in the body of the paper.

In this paper we investigate in a systematic way the solution structure of nonnegative solutions for coupled nonlinear Schrödinger systems in the all attractive regime. For all frequencies equal case we obtain results on Morse index of synchronized positive vector solutions and nonnegative semi-vector solutions as well as their kernel of linearized systems at these solutions. We establish a variational characterization of synchronized positive vector solutions and as applications we give a new existence result about positive vector solutions for the general systems with arbitrary frequencies. We also examine the synchronization phenomenon of positive vector solutions, and prove that if the interaction matrix has exactly one positive eigenvalue, any positive vector solution is synchronized. Finally we provide examples of domains for which synchronization of positive solutions fails to hold if the interaction matrix has at least two positive eigenvalues. Our results reveal the effect of the spectral information of the interaction matrix of the couplings and geometry of a domain on the solution structure.

For $(Q_t,P_t)$ governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a $\sigma$-strongly log-concave target $\pi(dq) = \frac{1}{Z}e^{-V(q)}dq$ as \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq e^{-\sqrt{\sigma} t}\operatorname{KL}(\operatorname{Law}(Q_{0}, P_{0})\, \|\, \Pi_0), \end{align*} where $\Pi_0$ denotes an appropriately selected reference measure. When $\pi$ is merely log-concave, the estimate \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq \frac{\tau^2}{t^2}\operatorname{KL}(\operatorname{Law}(Q_{\tau}, P_{\tau})\, \|\, \Pi_{\tau}) \end{align*} is derived, where $\Pi_\tau$ denotes another correspondingly chosen reference measure at time $\tau > 0$. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in $\mathbb{R}^d$. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.

We prove the sharp quantitative Faber–Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint–Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.

In this paper we establish a necessary and sufficient condition for the solvability of the Dirichlet problem in the unit ball $\mathbb{B}$ for the second order system $\partial_{\underline{x}} f\partial_{\underline{x}}=0$, where $\partial_{\underline{x}}$ stands for the Clifford-algebra valued Dirac operator in $\mathbb{R}^m$. We prove that the problem admits a solution whenever the boundary data belong to $C^1(\partial\mathbb{B})$. Conversely, we construct explicit counterexamples showing that solvability may fail if this smoothness assumption on the boundary is not satisfied. Moreover, in contrast with the standard commutative setting, we show how that solutions exhibit a regularity one order less than the boundary data, when the latter belong to $C^{k,\nu}(\partial\mathbb{B})$.

Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust–Hille constants governed by interaction order grow polynomially: if $p_d=2d/(d+1)$ and \[ \gamma_2=\frac12, \qquad \gamma_K=\frac{K\log(K-1)}{4(K-2)}\quad(K\ge3), \] then the $\ell^{p_d}$ norm of Fourier coefficients $\{\hat f(\alpha)\}$ is bounded by $L^\infty$ norm of $f$ multiplied by $C(K) d^{4\gamma_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5/2}$.

In these lecture notes, we review the recent results around the chain rule for the generalised KPZ equation and the geometric KPZ equation. We introduce the main ideas of Regularity Structures via a B-series type expansion and the two main combinatorial sets for encoding this expansion: decorated trees and multi-indices. Then, we explain how the chain rule was first understood via integration by parts and diagrammatic computations. For the generalised KPZ equation, a scalar-valued equation, we use multi-indices to derive a characterisation in terms of iterations of covariant derivatives. The same characterisation for decorated trees is more involved and requires the use of operadic and homological tools. We summarise the main arguments of this proof.

We establish a weak solution theory and a singular constitutive limit for a non variational Cahn Hilliard evolution on compact Riemannian manifolds, possibly with boundary. The model combines the classical double well chemical potential with an active correction concentrated in diffuse transition layers and sensitive to the sign of the Laplace Beltrami operator, together with a monotone anchoring mechanism toward a prescribed reference state. The active correction breaks the passive gradient flow structure and introduces a nonlinear dependence on second derivatives not identified by the natural weak compactness estimates. For square integrable initial data and sufficiently integrable reference data, we construct weak solutions in arbitrary dimension. A Galerkin level one sided comparison argument, using monotonicity of the classifier and anchoring law together with biharmonic coercivity, yields strong convergence of the approximate Laplacians and hence strong second order compactness. This identifies the nonlinear active term in an ordinary weak formulation, without a second derivative defect. In two dimensions we prove uniqueness and continuous dependence. The estimates depend only on the bound and monotonicity of the classifier, not on its slope, and are therefore uniform for increasingly steep arctangent classifiers. Their singular limit is governed by the maximal monotone sign graph acting on the Laplace Beltrami operator. In arbitrary dimension we obtain subsequential strong second order convergence to a weak solution of the resulting differential inclusion. In two dimensions the limiting state is unique, so the entire steep classifier family converges; uniqueness of the constitutive multiplier on the zero Laplacian set is not asserted. This is a constitutive steepness limit at fixed diffuse interface thickness, rather than a sharp interface limit.

We study the critical biharmonic equation \[ \Delta^2 u=u^{\frac{n+4}{n-4}}, \] on a complete, connected, and non-compact Riemannian manifold \((M^n,g)\) of dimension \(n\geq 5\) with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein's method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space \(\mathbb{R}^n\), and the solution is given by \[ u(x)= \frac{ \left[\lambda^2 n(n-4)(n^2-4)\right]^{\frac{n-4}{8}} } {\left(1+\lambda |x-x_0|^2\right)^{\frac{n-4}{2}}}, \qquad \lambda>0,\quad x_0\in\mathbb{R}^n . \]

Semilinear evolution equations are solved by means of the variation of constants formula under the action of smoothing linear operators and a finite number of nonlinearities operating between chosen Banach spaces of a given family. Admissible spaces of initial conditions are determined for the existence and uniqueness of a suitable notion of solution. Its maximal time of existence, short time and blow up time profiles, global extendibility and regularity are analyzed. For extrapolated fractional power scale of Banach spaces the fulfillment of the corresponding Cauchy problem is shown. Chosen applications to the modified viscous Cahn-Hilliard equation in R^N, the evolution of cell population of varying genotype and the semilinear Schrödinger equation are presented.

Let $L=-\Delta+V$ be a Schrödinger operator on $\mathbb{R}^n$, where $\Delta$ is the Laplacian and $V$ satisfies the reverse Hölder inequality ${\rm RH}_q$ for some $q>n/2$. In this paper, we study the behavior of the Littlewood–Paley operators $s_L$ and $S_L$, as well as the semigroup maximal operator $T^*_L$, on the space ${\rm CMO}_L(\mathbb{R}^n)$ associated with the Schrödinger operator $L$. It is known from previous work that these operators are bounded on ${\rm BMO}_L(\mathbb{R}^n)$. Our main result shows that they are, in fact, mappings from ${\rm CMO}_L(\mathbb{R}^n)$ into itself. To prove this, we develop several equivalent characterizations of ${\rm CMO}_L(\mathbb{R}^n)$ and employ a refined decomposition that partitions the parameter interval at $r_B\rho(x_B)$, instead of the customary $r_B^2$ or $\rho(x_B)^2$. The new strategy allows us to overcome a key technical obstacle that arises when applying existing methods to the ${\rm CMO}_L$ setting.

We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation associated with $\mathcal L_a=-\Delta+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^\infty$. For negative potentials in the global well-posedness range, set $\sigma=\frac12-\sqrt{\frac14+a}$. We obtain the free strong decay rate for $2<p<3/\sigma$ and the limiting Lorentz estimate from $L^{(3/\sigma)',1}$ to $L^{3/\sigma,\infty}$. The restriction $p<3/\sigma$ is sharp for strong Lebesgue decay. The proof combines a finite-interval bootstrap, nonlinear real interpolation on bounded energy sets, and endpoint Sobolev–Lorentz estimates adapted to $\mathcal L_a$.

We show the following equivalence of solutions to general fractional Helmholtz equations: $u$ is a solution to $$A^su=\lambda^su$$ for some fixed $0<s<2$ and some $\lambda>0$ if and only if $u$ is a solution to the same equation for all $s\neq0$. Furthermore, we show as well that $u$ equivalently solves the logarithmic Helmholtz equation $$\log(A)u=(\log\lambda)u.$$ Here, $A$ is a nonnegative, linear operator on a Banach space $X$. In particular, our results hold whenever $A=-L$, where $L$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$ satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian $(-\Delta)^s$ in $\mathbb{R}^n$. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace–Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.

We explore the regularity of energy maximizers for the Lagrangian of a periodic dispersion managed fiber optic at fixed intensity, on a torus of length $L$ and with vanishing average dispersion. We show that the Fourier coefficients decay at a polynomial rate, and then upgrade this to exponential decay, so that the maximizers are analytic in space for large enough $L$. In addition, by an asymptotic comparison to the optimizers on the real line, we prove that the solutions are non-trivial. We also consider a conjecture that the maximizer necessarily has an underlying symmetry inherent to both the energy functional and the resulting Euler-Lagrange equation. All of our results are supported with illustrative numerical experiments.

We introduce a cross-product Landau gradient that commutes with Gaussian mollification. For the hard-potential interaction kernels of the form $A(|v-v_*|)\sim \langle v-v_*\rangle^\gamma|v-v_*|^2$ with $\gamma\in(-\infty,1]$, this construction yields a variational characterisation of the gradient-flow structure of the spatially homogeneous Landau equation. The cross-product gradient induces the same gradient-flow geometry as that introduced by Carrillo, Delgadino, Desvillettes and Wu (2024), while providing a different representation of the Landau gradient. The main advantage of this formulation is that it requires only weighted $L^1$-control. We also discuss a similar variational characterisation for the GENERIC structure of the fuzzy Landau equations.

We prove that the gradient graph of any $W^{2,1}$ semi-convex viscosity solution to the three-dimensional subcritical special Lagrangian equation is area-minimizing and smooth. Furthermore, we prove the smoothness of $W^{2,1}\cap C^{1,1/3+}$ semi-convex viscosity solutions. This yields progress on the conjecture proposed by Wang–Yuan [WY13]. The corresponding results in higher dimensions are also established.

This paper establishes the fully discrete stochastic maximal $L^p$-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization $A_h$ of a general second-order elliptic operator $A=-\nabla \cdot a\nabla +b\cdot \nabla +c$ with Dirichlet boundary conditions on a smooth, bounded, convex domain in $\mathbb{R}^3$, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded $H^\infty$-calculus for the discrete spatial operator $A_h$, uniformly in the mesh size $h$. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations. As an application, we establish the pathwise uniform convergence of a fully discrete approximation of the three-dimensional stochastic Allen–Cahn equation with variable diffusion coefficients and multiplicative noise.

We investigate the existence and nonexistence of least energy solutions for a doubly critical Neumann problem with a critical absorption term. We consider a positive function $u$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^n$ with $n\ge 5$, satisfying $-\Delta u + \lambda u = u^{2^* - 1} -\alpha u^{2^\sharp-1}$ inside $\Omega$, with Neumann boundary condition $\nabla u\cdot \nu = u^{2^\sharp -1}$ on $\partial\Omega$, where $\lambda>0$, $\alpha\ge0$, $2^*=\frac{2n}{n-2}$ is the critical Sobolev exponent and $2^\sharp=\frac{2(n-1)}{n-2}$ denotes the critical trace exponent. The simultaneous presence of the interior and boundary critical exponents together with this critical absorption term creates a new interaction between competing concentration mechanisms. In particular, the absorption term has the same critical trace exponent as the boundary nonlinearity, but with the opposite sign and acting in the interior of $\Omega$. Consequently, it competes with the boundary mean curvature correction at the same asymptotic order in the energy expansion, leading to a sharp threshold phenomenon. Combining several analytic techniques and geometric tools, we prove the existence of a threshold value $\alpha_{0}=\alpha_{0}(\lambda,\Omega)\in(0,+\infty)$ such that the problem admits a least energy solution if $\alpha<\alpha_{0}$, and no least energy solution if $\alpha>\alpha_{0}$. Moreover, the problem also has a least energy solution at $\alpha =\alpha_0$ provided $\alpha_0> C(n)\max_{\partial \Omega} H$, where $C(n)$ is a positive constant depending only on $n$ and $H$ is the mean curvature on $\partial\Omega$.

In this paper, we study the low Mach number limit for a compressible two-fluid model with algebraic closure in the $d$-dimensional torus with $d \geq 2$. For large and ill-prepared initial data in critical Besov spaces, we prove that, provided that the Mach number is sufficiently small, the rescaled compressible two-fluid flow exists in critical Besov spaces for any finite time not exceeding the lifespan of the incompressible flow. Moreover, the rescaled compressible two-fluid flow converges to the incompressible Navier–Stokes flow as the Mach number tends to zero. The proof is based on a high-middle-low frequency analysis of the densities and velocity field, combined with a filtering technique involving wave operators. The main novelty is the derivation of new a priori estimates for the high-middle frequency part of the solution to the compressible two-fluid model, depending explicitly on time, the frequency parameter and the Mach number. To the best of our knowledge, this is the first work that proves (almost) global convergence for large and ill-prepared initial data in the low Mach number limit for compressible two-fluid model in critical framework.

We consider large time asymptotics of solutions to the damped Schrödinger equation with the nonlinear dissipation in the mass-subcritical case. We prove that the optimal $L^2$-decay rate of the nonlinear solution coincides with that of the corresponding linear solution even in the presence of nonlinear dissipation. Moreover, we give the optimal convergence rate for scattering for any power in the mass-subcritical regime.

Ultrafunctions are generalized functions defined on a non-Archimedean field extending R. They provide a framework for treating problems that fall outside the classical theory of distributions. In this paper we introduce an improved notion of ultrafunctions which refines the previous constructions. This new approach allows a more flexible functional framework than the one used in other papers and yields a more flexible functional setting. In particular, this approach allows us to prove the existence of ultrafunction solutions for several classes of ill-posed evolution problems arising in partial differential equations.

We study positive entire solutions of the flat constant $Q/R$-curvature equation in dimension $n\geq5$. We prove that the weak superharmonicity condition $-\Delta u\geq0$ already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants or standard bubbles. We also show that the weak superharmonicity condition is structurally necessary. Once it is removed, there is a one-parameter family of positive radial entire solutions near every positive constant. These solutions converge to a positive constant at infinity, while both $-\Delta u$ and $-\Delta(u^{(n-2)/(n-4)})$ change sign infinitely many times.

We study the existence of nontrivial solutions to nonlocal semilinear diffusion equations of the form \[ a_0(x)u(x)-\int_\Omega k(x,y)\big(u(y)-u(x)\big)\,d\mu(y)=F(x,u), \quad u\geq 0, \] on a general measure space $(\Omega,M,\mu)$, where the kernel $k$ is nonnegative and symmetric. In this framework, a major challenge is establishing the strict positivity of solutions. To this end, we provide a novel complete characterization of the strong maximum principle in terms of a connectivity structure induced by the kernel, which leads to a decomposition of the domain into independent components. We also develop the method of sub- and supersolutions in this setting, obtaining existence, uniqueness, and nonexistence results, as well as a regularity result that ensures the continuity of solutions under suitable assumptions. The main interest of working within an abstract measure space lies in its rich applicability, as it unifies and extends a wide variety of frameworks, including discrete, continuous, and hybrid models, or systems interacting across different dimensions. Finally, numerical experiments are presented to illustrate the potential of our theoretical results.

We prove existence and uniqueness for the capillary $L_p$ dual Minkowski problem for $p>q$ and contact angle $\theta\in (0,\frac{\pi}{2})$ in $\mathbb{R}^{n+1}$, $n\geq 3$. We reduce it to a Monge-Ampère type equation with a Robin boundary condition on the unit spherical cap, by building a new auxiliary function to obtain the $C^2$ estimate for $p>q$ in $\mathbb{R}^{n+1}$ ($n\geq 3$), we then prove that there exists a unique smooth solution that solves this problem provided $\theta\in (0, \frac{\pi}{2})$.